Question: Find the number of degrees in the measure of angle $x$.

[asy]
import markers;
size (5cm,5cm);
pair A,B,C,D,F,H;

A=(0,0);
B=(5,0);
C=(9,0);
D=(3.8,7);

F=(2.3,7.2);
H=(5.3,7.2);

draw((4.2,6.1){up}..{right}(5.3,7.2));
draw((3.6,6.1){up}..{left}(2.3,7.2));

draw (A--B--C--D--A);
draw (B--D);

markangle(n=1,radius=8,C,B,D,marker(stickframe(n=0),true));

label ("$x^\circ$", shift(1.3,0.65)*A);
label ("$108^\circ$", shift(1.2,1)*B);
label ("$26^\circ$", F,W);
label ("$23^\circ$",H,E);

[/asy]
[asy]
import markers;
size (5cm,5cm);
pair A,B,C,D,F,H;

A=(0,0);
B=(5,0);
C=(9,0);
D=(3.8,7);

F=(2.3,7.2);
H=(5.3,7.2);

draw((4.2,6.1){up}..{right}(5.3,7.2));
draw((3.6,6.1){up}..{left}(2.3,7.2));

draw (A--B--C--D--A);
draw (B--D);

markangle(n=1,radius=8,C,B,D,marker(stickframe(n=0),true));

label ("$x^\circ$", shift(1.3,0.65)*A);
label ("$108^\circ$", shift(1.2,1)*B);
label ("$26^\circ$", F,W);
label ("$23^\circ$",H,E);
label("$A$",A,S);
label("$B$",B,S);
label("$C$",C,S);
label("$D$",D,N);
[/asy]

Since $\angle ABC$ is a straight angle, we have $\angle ABD = 180^\circ - 108^\circ = 72^\circ$.  From triangle $ABD$, we have  \[26^\circ + 72^\circ + x = 180^\circ,\]  so $98^\circ + x = 180^\circ$ and $x = \boxed{82^\circ}$.